| L(s) = 1 | + 0.430·2-s − 0.376·3-s − 1.81·4-s − 3.33·5-s − 0.162·6-s − 1.83·7-s − 1.64·8-s − 2.85·9-s − 1.43·10-s − 0.338·11-s + 0.684·12-s − 13-s − 0.790·14-s + 1.25·15-s + 2.92·16-s + 3.04·17-s − 1.22·18-s − 4.00·19-s + 6.05·20-s + 0.692·21-s − 0.145·22-s − 2.44·23-s + 0.618·24-s + 6.12·25-s − 0.430·26-s + 2.20·27-s + 3.33·28-s + ⋯ |
| L(s) = 1 | + 0.304·2-s − 0.217·3-s − 0.907·4-s − 1.49·5-s − 0.0662·6-s − 0.694·7-s − 0.580·8-s − 0.952·9-s − 0.453·10-s − 0.102·11-s + 0.197·12-s − 0.277·13-s − 0.211·14-s + 0.324·15-s + 0.730·16-s + 0.738·17-s − 0.289·18-s − 0.917·19-s + 1.35·20-s + 0.151·21-s − 0.0310·22-s − 0.509·23-s + 0.126·24-s + 1.22·25-s − 0.0843·26-s + 0.424·27-s + 0.630·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3878377679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3878377679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.430T + 2T^{2} \) |
| 3 | \( 1 + 0.376T + 3T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + 0.338T + 11T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 19 | \( 1 + 4.00T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 - 6.20T + 41T^{2} \) |
| 43 | \( 1 - 6.67T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 - 6.05T + 83T^{2} \) |
| 89 | \( 1 + 4.98T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.406046130310817364832064529239, −8.849178053192103961697016734546, −7.942789168780932555338287731787, −7.39473651319831294006185496558, −6.05929850931615808425294593268, −5.46117226804035658712037831629, −4.24921847479126307245240077834, −3.77254288467493844123819096809, −2.78736733246373792004270829278, −0.41701996988200336763224928406,
0.41701996988200336763224928406, 2.78736733246373792004270829278, 3.77254288467493844123819096809, 4.24921847479126307245240077834, 5.46117226804035658712037831629, 6.05929850931615808425294593268, 7.39473651319831294006185496558, 7.942789168780932555338287731787, 8.849178053192103961697016734546, 9.406046130310817364832064529239
